Work problem number 17 from section 4.4 of your book (page 194) using the revenue function R(q)=630q and the cost function C(q)=9500+2q2. Round your answers to the nearest whole number.
a) At what quantity is profit maximized?
q =
b) What is the total profit at this production level?
profit = $
profit P(q) = R(q) - C(q)
Now just find where dP/dq = 0
To find the quantity at which profit is maximized, we need to first determine the profit function. The profit function is given by the equation:
P(q) = R(q) - C(q)
where R(q) is the revenue function and C(q) is the cost function.
Given:
R(q) = 630q
C(q) = 9500 + 2q^2
We can substitute these functions into the profit function equation:
P(q) = 630q - (9500 + 2q^2)
P(q) = 630q - 9500 - 2q^2
Now, to find the quantity at which profit is maximized, we find the derivative of the profit function with respect to q and set it equal to zero:
P'(q) = 630 - 4q
Setting P'(q) = 0, we have:
630 - 4q = 0
4q = 630
q = 157.5
Since the quantity should be rounded to the nearest whole number, the quantity at which profit is maximized is q = 157.
To determine the total profit at this production level, we substitute the value of q into the profit function:
P(q) = 630q - 9500 - 2q^2
P(157) = 630(157) - 9500 - 2(157)^2
P(157) = 98810 - 9500 - 2(24649)
P(157) = 98810 - 9500 - 49298
P(157) = 44512
Therefore, the total profit at this production level is $44,512.
To solve this problem, we need to find the quantity of the production level that maximizes profit and calculate the total profit at that level.
First, let's write down the profit function, which is the difference between the revenue and the cost functions:
Profit (P) = Revenue (R) - Cost (C)
Given:
Revenue function R(q) = 630q
Cost function C(q) = 9500 + 2q^2
Now, let's substitute the revenue and cost functions into the profit function:
P(q) = R(q) - C(q)
P(q) = 630q - (9500 + 2q^2)
To find the quantity (q) that maximizes profit, we need to find the derivative of the profit function and set it equal to zero. So we differentiate the profit function with respect to q:
P'(q) = 630 - 4q
Now, set P'(q) equal to zero and solve for q:
630 - 4q = 0
4q = 630
q = 630/4
q = 157.5
Since the quantity must be a whole number, we round q to the nearest whole number:
q = 158
So at a production level of 158, profit is maximized.
Next, to find the total profit at this level, substitute q = 158 into the profit function P(q):
P(158) = 630(158) - (9500 + 2(158^2))
P(158) = 99540 - (9500 + 2(24964))
P(158) = 99540 - (9500 + 49928)
P(158) = 99540 - 59428
P(158) = 40112
Therefore, at a production level of 158, the total profit is $40,112.